Kinematics

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Displacement,
Velocity & Accel.
Motion
Graphs
Deriving
Equations
Solving
Problems
Free Fall
Experiment
Projectile
Motion

Displacement, Velocity & Acceleration

Distance and Displacement

Distance:
  • Distance is a measure of how far an object travels.
  • It is a scalar quantity with magnitude only.
300 m RACE START FINISH (300m mark)

The athletes run a total distance of 300 m.

Displacement:
  • Displacement is a measure of how far something is from its starting position, along with its direction.
  • In other words, it is the change in position.
  • It is a vector quantity with both magnitude and direction.

Total Distance vs Total Displacement Example:

  • Consider athletes running a 300 m race on a 400 m track.
  • Total Distance = 300 m.
  • Final Displacement = 100 m to the right (if they end 100m from start).
  • If they ran full 400 m, final displacement would be zero.

Speed and Velocity

Speed

  • The distance traveled every second.
  • Scalar quantity (magnitude only).
  • average speed =
    total distance time taken
  • SI Units: meters per second (m s-1).

Velocity

  • The rate of change of displacement.
  • Vector quantity (magnitude + direction).
  • Speed in a given direction.
  • vavg =
    Δx Δt
  • (if u and v known): vavg =
    u + v 2
Examiner Tip: Velocity is speed in a given direction, but average velocity is NOT average speed in a given direction. Speed uses distance; Velocity uses displacement.

Acceleration

Defined as: The rate of change of velocity.

It is a vector quantity measured in m s-2.

a =
Δv Δt
=
v - u t

Where v = final velocity, u = initial velocity.

Worked Example: Professor's Walk

A professor walks path ABCDA around her garden.

A B C D 15 km 9 km

Calculate Distance: Sum of all sides: 15 + 9 + 15 + 9 = 48 km.

Calculate Displacement: She travels back to A. Distance from original position is 0. Displacement = 0 km.

Quiz: Displacement & Velocity

Motion Graphs

Distance-Time Graphs

  • Slope equals speed.
  • y-intercept equals initial position.
  • Straight (diagonal) line = constant speed.
  • Curved line = acceleration.
  • Slope always zero or positive (distance is scalar).
  • Horizontal line = State of rest.

Displacement-Time Graphs

  • Slope equals velocity.
  • Positive slope = motion in positive direction.
  • Negative slope = motion in negative direction.
Time Disp

Constant Velocity

Increasing Velocity

Velocity-Time Graphs

  • Slope equals acceleration.
  • Area under the curve equals displacement.
  • Straight line = uniform acceleration.
  • Horizontal line = constant velocity (zero acceleration).

Const Velocity (a=0)

Const Acceleration

Increasing Accel.

Worked Example: Area under Graph

To find displacement: Split area into shapes (triangles/rectangles).

Area of triangle = 12 x base x height

Area of rectangle = base x height

Quiz: Motion Graphs

Deriving Kinematic Equations

Variables (SUVAT):

s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time

1. Deriving v = u + at

From the gradient of a velocity-time graph:

a =
v - ut

Rearrange: at = v - u → v = u + at

2. Deriving s = u + v2t

Displacement is the area under v-t graph.

Average velocity is halfway between u and v.

s =
u + v2
t

3. Deriving s = ut + 12at2

Area under graph = Rectangle (ut) + Triangle (12(v-u)t).

Since (v-u) = at:

s = ut +
12
at2

4. Deriving v2 = u2 + 2as

Substitute t = (v-u)/a into equation 2.

v2 = u2 + 2as
Important: Only equations 3 and 4 are usually on the data sheet. Memorize all!

Quiz: Derivations

Solving Problems (SUVAT)

Step-by-Step Method

  1. Write out variables known and unknown (s, u, v, a, t).
  2. Deduce implicit values (e.g., "starts at rest" → u=0; "gravity" → a=9.81).
  3. Choose the equation that contains the quantities you have.
  4. Convert units to SI units.
  5. Rearrange and solve.

Worked Example: Train Stopper

Train speed 50 m s-1. Must pass marker 2 at 10 m s-1. Time limit 20 s.

Knowns: u = 50, v = 10, t = 20, s = ?

Equation: s = u + v2 t

Calculation: s = 50 + 102 x 20 = 600 m

Quiz: Solving Problems

Acceleration of Free Fall

Aim

Calculate 'g' by measuring time for a ball-bearing to fall a distance.

  • Independent: Height (h)
  • Dependent: Time (t)

Apparatus Setup

Gate 1 Gate 2

Analysis

Use equation: s = ut + 12at2

Rearranged for straight line graph: 2h/t = gt + 2u

  • Plot 2h/t (y-axis) vs t (x-axis).
  • Gradient = g.
  • Y-intercept = 2u.

Quiz: Experiments

Projectile Motion

Key Concepts

Vertical and horizontal components are independent.

u Range (R) Max Height (H)

Horizontal (→)

a = 0

ux = u cos θ

Range R = (u cos θ)t

Vertical (↑)

a = -g

uy = u sin θ

Use SUVAT

Equations

Time of Flight: t = 2u sin θg

Max Height: H = (u sin θ)22g

Range: R = u2 sin 2θg

Worked Example: Motorcycle Jump

Takes off horizontally 1.25m high. Lands 10m away. What was speed?

Step 1: Vertical (Find Time)

s = 1.25, u = 0, a = 9.81

s = 0.5gt2 → t = 2s/g = 0.5 s

Step 2: Horizontal (Find Speed)

s = 10, t = 0.5, a = 0

u = s/t = 10/0.5 = 20 m s-1

Quiz: Projectiles

Practice Questions

Question 1 of X